Joel H. Shapiro

Composition operators and classical function theory

The study of composition operators forges links between fundamental properties of linear operators and results from the classical theory of analytic functions. This book provides a self-contained introduction to both the subject and its function-theoretic underpinnings. The development is geometrically motivated, and accessible to anyone who has studied basic graduate-level real and complex analysis. The work explores how operator-theoretic issues such as boundedness, compactness and cyclicity evolve. Each of these classical topics is developed fully, and particular attention is paid to their common geometric heritage as descendants of the Schwarz Lemma.

Operators with dense, invariant, cyclic vector manifolds☆

Abstract We study a class of Banach space operators patterned after the weighted backward shifts on Hilbert space, and show that any non-scalar operator in the commutant of one of these “generalized backward shifts” has a dense, invariant linear manifold whose non-zero members are cyclic vectors. Under appropriate extra hypotheses on the commuting operator, stronger forms of cyclicity are possible, the most extreme being hypercyclicity (density of an orbit). Motivated by these results, we examine the cyclic behavior of two seemingly unrelated classes of operators: adjoint multiplications on Hilbert spaces of holomorphic functions, and differential operators on the Frechet space of entire functions. We show that each of these operators (other than the scalar multiples of the identity) possesses a dense, invariant linear submanifold each of whose non-zero elements is hypercyclic. Finally, we explore some connections with dynamics; many of the hypercyclic operators discussed here are, in at least one of the commonly accepted senses of the word, “chaotic.”

Universal vectors for operators on spaces of holomorphic functions

A vector x in a linear topological space X is called universal for a linear operator T on X if the orbit {Tnx: n > 0} is dense in X. Our main result gives conditions on T and X which guarantee that T will have universal vectors. It applies to the operators of differentiation and translation on the space of entire functions, where it makes contact with Polyas theory of final sets; and also to backward shifts and related operators on various Hilbert and Banach spaces.

Angular derivatives and compact composition operators on the Hardy and Bergman spaces

Let U denote the open unit disc of the complex plane, and φ a holomorphic function taking U into itself. In this paper we study the linear composition operator C φ defined by C φ f = f o φ for f holomorphic on U . Our goal is to determine, in terms of geometric properties of φ , when C φ is a compact operator on the Hardy and Bergman spaces of φ . For Bergman spaces we solve the problem completely in terms of the angular derivative of φ , and for a slightly restricted class of φ (which includes the univalent ones) we obtain the same solution for the Hardy spaces H p (0 p

CYCLIC PHENOMENA FOR COMPOSITION OPERATORS

Introduction Preliminaries Linear-fractional composition operators Linear-fractional models The hyperbolic and parabolic models Cyclicity: Parabolic nonautomorphism case Endnotes References.

Compact composition operators on

The composition operator induced by a holomorphic self-map of the unit disc is compact on L1 of the unit circle if and only if it is compact on the Hardy space H2 of the disc. This answers a question posed by Donald Sarason: it proves that Sarasons integral condition characterizing compactness on Li is equivalent to the asymptotic condition on the Nevanlinna counting function which characterizes compactness on H2 .

Hardy spaces that support no compact composition operators

We consider, for G a simply connected domain and 0<p<∞, the Hardy space Hp(G) formed by fixing a Riemann map τ of the unit disc onto G, and demanding of functions F holomorphic on G that the integrals of |F|p over the curves τ({|z|=r}) be bounded for 0<r<1. The resulting space is usually not the one obtained from the classical Hardy space of the unit disc by conformal mapping. This is reflected in our Main Theorem: Hp(G) supports compact composition operators if and only if ∂G has finite one-dimensional Hausdorff measure. Our work is inspired by an earlier result of Matache (Proc. Amer. Math. Soc. 127 (1999) 1483), who showed that the Hp spaces of half-planes support no compact composition operators. Our methods provide a lower bound for the essential spectral radius which shows that the same result holds with “compact” replaced by “Riesz.” We prove similar results for Bergman spaces, with the Hardy-space condition “∂G has finite Hausdorff 1-measure” replaced by “G has finite area.” Finally, we characterize those domains G for which every composition operator on either the Hardy or the Bergman spaces is bounded.

Which linear-fractional composition operators are essentially normal?☆

Abstract We characterize the essentially normal composition operators induced on the Hardy space H2 by linear-fractional maps; they are either compact, normal, or (the nontrivial case) induced by parabolic nonautomorphisms. These parabolic maps induce the first known examples of nontrivially essentially normal composition operators. In addition, we characterize those linear-fractionally induced composition operators on H2 that are essentially self-adjoint, and present a number of results for composition operators induced by maps that are not linear-fractional.

On the Toeplitzness of Composition Operators

Composition operators on H 2 cannot–except trivially–be Toeplitz, or even ‘Toeplitz plus compact’. However there are natural ways in which they can be ‘asymptotically Toeplitz’. We show here that the study of such phenomena leads to surprising results and interesting open problems. †Dedicated to Professor Peter L. Duren on the occasion of his 70th birthday.

When is zero in the numerical range of a composition operator

AbstractWe work on the Hardy spaceH2 of the open unit disc